Solve for $x$ and $y$ using elimination. $\begin{align*}-6x+6y &= -6 \\ -2x-7y &= -3\end{align*}$
Solution: We can eliminate $x$ when its corresponding coefficients are negative inverses. Recalling our knowledge of least common multiples, multiply the top equation by $-1$ and the bottom equation by $3$ $\begin{align*}6x-6y &= 6\\ -6x-21y &= -9\end{align*}$ Add the top and bottom equations. $-27y = -3$ Divide both sides by $-27$ and reduce as necessary. $y = \dfrac{1}{9}$ Substitute $\dfrac{1}{9}$ for $y$ in the top equation. $-6x+6( \dfrac{1}{9}) = -6$ $-6x+\dfrac{2}{3} = -6$ $-6x = -\dfrac{20}{3}$ $x = \dfrac{10}{9}$ The solution is $\enspace x = \dfrac{10}{9}, \enspace y = \dfrac{1}{9}$.